Question on Twin Paradox

I have a question about the Twin Paradox. I don't know if I'm right or wrong, but thats why I figured I would come here to ask.

The way I understand things is that the twin on earth's clock would end up being faster than the twin's clock out in deep space if the twin out in space was approaching the speed of light. Wouldn't the twin out in deep space be more massive due to traveling at the speed of light since as a object with mass approaches the speed of light it mass increases therefore never letting it exceed the speed of light? So if he is getting more massive, wouldn't he then be more effected by gravity and therefore his clock would speed up due to the gravitational force?

Like I said I don't know if these have already been stated, but it was something that made me think this morning.

In the twin paradox, we assume there's no gravity (like we assume spherical cows).

The twin paradox is a "standard scenario" in the theory of special relativity, which doesn't work when there's gravity.

The general theory of relativity is required when there's gravity, and it is within that theory that gravity affects time. But I haven't thought about what happens to the twin paradox within that theory.

Just a general consideration. In the general theory of relativity, clocks in a strong gravitational field tick more slowly. Considering the effect of the earth's gravity, the clock on the earth will tick more slowly than the spaceship clock, counteracting the "standard scenario" time dilation effect, as you intuited. It's not intuitively clear to me which of the many time dilation effects will win out.

One experiment which compared special and general relativistic time dilations was the Hafele-Keating experiment. Both time dilations are also taken into account in keeping GPS satellite clocks synchronized with those on earth. Apparently, "For a low earth orbiter such as the Space Shuttle, the velocity is so great that slowing due to time dilation is the dominant effect, while for a GPS satellite clock, the gravitational blueshift is greater. (http://relativity.livingreviews.org/Articles/lrr-2003-1/)"